Taylor polynomial
of degree , for a function f that is n times differentiable at x=x_0
The polynomial P_n(x)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k. The values of the Taylor polynomial and of its derivatives up to order n inclusive at the point x=x_0 coincide with the values of the function and of its corresponding derivatives at the same point: f^{(k)}(x_0)=P_n^{(k)}(x_0),\quad k=0,\dotsc,n. The Taylor polynomial is the best polynomial approximation of the function f as x\to x_0, in the sense that
f(x)-P_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0, | (*) |
and if some polynomial Q_n(x) of degree not exceeding n has the property that f(x)-Q_n(x)=o\left((x-x_0)^m\right),\quad x\to x_0, where m\ge n, then it coincides with the Taylor polynomial P_n(x). In other words, the polynomial having the property (*) is unique.
If at least one of the derivatives f^{(k)}(x), k=0,\dotsc,n, is not equal to 0 at the point x_0, then the Taylor polynomial is the principal part of the Taylor formula.
Comments
For references see Taylor formula.
Taylor polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Taylor_polynomial&oldid=29526